Taylor Series

Question 1

Derivation

Answer 1

Assume function can be written as infinite power series centered around x=a. + Function is infinitely differentiable at x=a

f(x) = Σcₙ(x-a)ⁿ = c₀ + c₁(x-a) c₂(x-a)² …
at x=a, c₀ = a
f(a) = c₀
f’(a) = c₁ + 2c₂(x-a) (x-a terms to 0)
f’‘(a) = 2.1.c₂

At x=a, cₙ = fⁿ (x)/n!
sub into power series

f(x) = f(a) + f’(a)(x-a) +f’(a)(x-a)² … fⁿ(x)(x-a)ⁿ . 1/n!
(for a function f(x) centered at x=a)

replace x w (x+a) to get in powers of x

Valid for values of x s.t. infinite series converges

maclaurin expansion of lnx not possible as f’(x) = 1/x (1/0)

Question 2

When subbing in for taylor approximations

Answer 2

If ask for sin(40degrees) make sure u convert to radians

Ev differentiation/integration is in radians

Question 3

Generalising Taylor expansion

Answer 3

f(a) + sigma[ f(kofa)(x-a)^k/k!]

from k = 1 bc 1st deriv to inf

Question 4

Generalising nth deriv of lnx

Answer 4

1st is 1/x
-1/x
2.1/x
-3.2.1/x

-1^(n-1)((k-1)!/x)